Flow Turbulence Combust (2006) 77: 277–303
DOI 10.1007/s10494-006-9047-1
Transition Modelling for General Purpose CFD Codes
F. R. Menter · R. Langtry · S. Völker
Accepted: 21 March 2006 / Published online: 18 August 2006
© Springer Science + Business Media B.V. 2006
Abstract The paper addresses modelling concepts based on the RANS equations
for laminar-turbulent transition prediction in general-purpose CFD codes. Available models are reviewed, with emphasis on their compatibility with modern CFD
methods. Requirements for engineering transition models suitable for industrial
CFD codes are specified. A new concept for transition modeling is introduced. It
is based on the combination of experimental correlations with locally formulated
transport equations. The concept is termed LCTM – Local Correlation-based Transition Model. An LCTM model, which satisfies most of the specified requirements
is described, including results for a variety of different complex applications. An
incremental approach was used to validate the model, first on 2D flat plates and
airfoils and then on to progressively more complicated test cases such as a threeelement flap, a 3D transonic wing and a full helicopter configuration. In all cases good
agreement with the available experimental data was observed. The authors believe
that the current formulation is a significant step forward in engineering transition
modeling, as it allows the combination of transition correlations with general purpose
CFD codes. There is a strong potential that the model will allow the 1st order effects
of transition to be included in everyday industrial CFD simulations.
Key words laminar-turbulent transition · intermittency · local formulation ·
turbulence modelling · transport equation · SST model · LCTM
F. R. Menter (B) · R. Langtry
Software Development Dept., ANSYS-CFX Germany, 83624 Otterfing, Germany
e-mail: [email protected]
S. Völker
GE Global Research, One Research Circle, Niskayuna, NY 12309, USA
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1. Introduction
In the last decades, significant progress has been made in the development of reliable
turbulence models, which allow the accurate simulation of a wide range of fully
turbulent engineering flows. The efforts by different groups have resulted in a
spectrum of models, which can be used for different applications, while balancing
the accuracy requirements and the computational resources available to a CFD user.
However, the important effect of laminar-turbulent transition is not included in the
large majority of today’s engineering CFD simulations. The reason is that transition
modelling does not offer the same wide spectrum of CFD-compatible model formulations as available for turbulent flows, although a large body of publications is
available on the subject. There are several reasons for this unsatisfactory situation.
The first is that transition occurs through different mechanisms in different
applications. In aerodynamic flows, transition is typically a result of a flow instability (Tollmien–Schlichting waves or cross-flow instability), where the resulting
exponential growth eventually results in a nonlinear breakdown to turbulence. In
turbomachinery applications, the main transition mechanism is bypass transition [11,
18] imposed on the boundary layer by high levels of turbulence in the free stream,
coming from the upstream blade rows. Another important transition mechanism is
separation-induced transition [12], where a laminar boundary layer separates under
the influence of a pressure gradient and transition develops within the separated
shear layer (which may or may not reattach). Finally, an already turbulent boundary
layer can relaminarise under a strong favourable pressure gradient [10]. While the
importance of transition phenomena for aerodynamic and heat transfer simulations
is widely accepted, it is difficult to include all of these effects in a single model.
The second complication arises from the fact that the conventional (RANS)
averaging procedures do not lend themselves easily to the description of transitional
flows, where both, linear and nonlinear effects are relevant. RANS averaging eliminates the effects of linear disturbance growth and is therefore difficult to apply
to the transition process. While methods based on the stability equations, like the
en method of Smith and Gamberoni [27] and van Ingen [36] avoid this limitation,
they are not compatible with general-purpose CFD methods as typically applied in
complex geometries. The reason is that these methods require a priori knowledge
of the geometry and the grid topology. In addition, they involve numerous nonlocal
operations, which are not easily implemented into today’s CFD methods [31]. This
is not to argue against these models, which are an essential part of the desired
‘spectrum’ of transition models required for the vastly different application areas and
accuracy requirements. However, much like in turbulence modelling, it is important
to develop engineering models, which can be applied in the day-to-day operation by
design engineers on varying geometries.
Closer inspection shows that hardly any of the current transition models are CFDcompatible. Most formulations suffer from nonlocal operations, which cannot be
carried out (with reasonable effort) in a general-purpose CFD code. It has to be
understood that modern CFD codes do not provide the infrastructure of computing
integral boundary layer parameters, or allow the integration of quantities along the
direction of external streamlines. Even if structured boundary layer grids are used
(typically hexahedra or prism layers), the codes are based on data structures for
unstructured meshes. The information on a body-normal grid direction is therefore
not easily available. In addition, industrial CFD simulations are carried out on
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parallel computers using a domain decomposition methodology. This means in
the most general case that boundary layers are typically split and computed on
different processors, prohibiting any search or integration algorithms. Furthermore,
for general purpose CFD applications, the grid topology relative to the surfaces is not
known a priori, as the user has the freedom to freely choose both, geometry and grid
topologies. The main requirements for a fully CFD-compatible transition model are
therefore:
1.
2.
3.
4.
Allow the calibrated prediction of the onset and the length of transition.
Allow the inclusion of the different transition mechanisms.
Be formulated locally (no search or line-integration operations).
Avoid multiple solutions (same solution for initially laminar or turbulent boundary layer).
5. Do not affect the underlying turbulence model in fully turbulent regimes.
6. Allow a robust integration with similar convergence as underlying turbulence
model.
7. Be formulated independent of the coordinate system.
Considering the main classes of engineering transition models (stability analysis,
correlation based models, low-Re models) one finds that most of these methods lack
one or the other of the above requirements.
The only formulations, which have historically been compatible with CFD methods, are low-Re models [3, 19]. However, they typically suffer from the close
interaction of the transition capability and the viscous sublayer modelling, which
prevents an independent calibration of both phenomena [22]. In addition, low-Re
models can at best be expected to simulate bypass transition, which is dominated by
diffusion effects. From a global perspective (without accounting for the differences
between different models in the same group), standard low-Re models rely on the
ability of the wall damping terms to also capture some of the effects of transition.
Realistically, it would be surprising if models calibrated for viscous sublayer damping
would faithfully reproduce the many effects of transitional flows. It is understandable
that models using damping functions based on the turbulent Reynolds number have
some transition characteristics. Nevertheless, the effect is best described as ‘pseudotransition,’ as it was never actually built into the model. However, there are several
models, where transition prediction was considered during model calibration [8,
37, 38]. It is interesting to note that several of these models use the strain-rate
(or vorticity-rate) Reynolds number ReV as an indicator for estimating the state of
the laminar boundary layer. Nevertheless, these model formulations are based on a
close connection of the sublayer and the transition calibration. Re-calibration of one
model functionality also changes the performance of the other. It is therefore not
possible to introduce additional experimental information, without a substantial reformulation of the entire model. In most cases this operation can only be performed
reliably by the model developer (or experts on model formulation). More complex
models for transitional flows based on solving additional transport equations for
either intermittency or pseudo laminar fluctuations have been developed by Steelant
and Dick [29] and Lardeau et al. [9]. These models do however require a separate
transition onset criteria, which is typically not formulated locally.
The engineering alternative to low-Re models are correlation-based formulations
like those of Abu–Ghannam and Shaw [1], Mayle [10] and Suzen et al. [32]. They
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typically correlate the transitional (momentum thickness) Reynolds number to local
free-stream conditions, like turbulence intensity and pressure gradient. These models
allow for an easy calibration and are often sufficiently accurate for capturing the
major effects. In addition, correlations can be developed for the different transition
mechanisms, ranging from bypass to natural transition to crossflow instability to
separation induced transition. The main shortcoming of these models lies in their
inherently nonlocal formulation. They typically require information on the integral
thickness of the boundary layer and the state of the flow outside the boundary layer.
While these models have been used successfully in special-purpose turbomachinery
codes, the nonlocal operations have precluded their implementation into generalpurpose CFD methods.
Transition simulations based on linear stability analysis, like the en method, are the
lowest closure level, where the actual instability of the flow is simulated. However,
even the en method is not free of empiricism, as the n-factor is not universal and
depends on the wind tunnel or free-stream environment. The main obstacle to the use
of the en model is however that the required infrastructure is typically very complex.
The stability analysis is often based on velocity profiles obtained from highly resolved
boundary layer codes, which are coupled to the pressure distribution of the RANS
solver. The output of the boundary layer method is then transferred to a stability
method, which then provides information back to the turbulence model in the RANS
solver. The complexity of this set-up is mainly justified for special applications where
the flow is designed to remain close to the stability limit for drag reduction, like
laminar airplane wing design.
DNS and LES are suitable tools for transition prediction (e.g. [17, 39]), although
even there, the proper specification of the external disturbance level and structure
poses substantial challenges. These methods are far too costly for engineering applications, and are currently used mainly as research tools and substitutes for controlled
experiments.
Despite its complexity, transition should not be viewed as outside the range of
RANS methods. In many applications, transition is enforced within a narrow area of
the flow by strong geometric disturbances, pressure gradients and/or flow separation.
Even relatively simple models can capture these effects with sufficient engineering
accuracy. The challenge to a proper engineering model is therefore mainly the
formulation of models, which are suitable for implementation into a general RANS
environment.
The present authors have recently developed a correlation-based transition model,
built on transport equations, using only local variables. The concept is termed LCTM
– Local Correlation-based Transition Model. The central idea behind this concept
has been described in a model by Menter et al. [15]. The major numerical and
modelling deficiencies associated with that prototype model have been eliminated by
Menter et al. [16] and a wide range of turbomachinery-related flow problems has been
computed by Langtry et al. [6]. The model has since been extended to aerodynamic
flows [7] and is now run within the software package CFX-5, as well as the GE
in-house code Tacoma on numerous industrial applications. The model satisfies all
requirements given above, except for the last one – coordinate independence. This
is a consequence of the fact that transition correlations are based on non-Galilean
invariant parameters, like the turbulence intensity Tu. As boundary layer transition
is always relative to walls, this is only an issue if multiple moving walls exist in a single
computational domain. Efforts are underway for eliminating this restriction.
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The model given in Menter et al. [16] and Langtry et al. [6] has been developed
in a joint project between GE Global Research, ANSYS-CFX and the University of
Kentucky. The model consists of two components. The first is the generic infrastructure provided by two transport equations, which link the CFD code to experimental
correlations. The second component are the correlations themselves. The innovation
lies in the generic infrastructure, which allows a direct coupling of general purpose
CFD method with experimental transition data. The entire infrastructure of the
formulation is given in Menter et al. [16] and will be repeated below. However, the
model correlations were partly built on internal data and are not in the public domain.
As the interfaces for the transitional correlations are clearly defined, other groups can
use their own correlations as available for their application.
2. Strain-Rate Reynolds Number
Instead of using the momentum thickness Reynolds number to trigger the onset
of transition, the current model is based on the strain-rate (or vorticity) Reynolds
number, Rev , [15, 35]:
Rev =
ρy2
S
µ
(1)
where y is the distance from the nearest wall, ρ is the density, µ is the dynamic
viscosity and S is the absolute value of the strain rate. Since the strain-rate Reynolds
number depends only on density, viscosity, wall distance and the strain-rate (some
formulations use the vorticity) it is a local property and can be easily computed at
each grid point in an unstructured, parallel Navier–Stokes code.
A scaled profile of the strain-rate Reynolds number is shown in Figure 1 for
a Blasius boundary layer. The scaling is chosen in order to have a maximum of
one inside the boundary layer and is achieved by dividing the strain-rate Reynolds
number profile from the Blasius solution by the corresponding momentum thickness
Reynolds number and a constant of 2.193. In other words, the maximum of the profile
Figure 1 Scaled strain-rate
Reynolds number (Rev ) profile
in a Blasius boundary layer.
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Figure 2 Relative error
between the maximum value
of strain-rate Reynolds
number (Rev ) and the
momentum thickness
Reynolds number (Reθ ) as a
function of boundary layer
shape factor (H).
is proportional to the momentum thickness Reynolds number and can therefore be
related to the transition correlations [15] as follows:
Reθ =
max(Rev )
2.193
(2)
Based on this observation, a general framework can be built, which can serve as
an environment for local correlation-based transition models (LCTM).
When the laminar boundary layer is subjected to strong pressure gradients,
the relationship between momentum thickness and strain-rate Reynolds number
described by Equation (2) changes due to the change in the shape of the profile. The
relative difference between momentum thickness and strain-rate Reynolds number,
as a function of shape factor (H), is shown in Figure 2. For moderate pressure
gradients (2.3 < H < 2.9) the difference between the actual momentum thickness
Reynolds number and the maximum of the strain-rate Reynolds number is less
than 10%. Based on boundary layer analysis a shape factor of 2.3 corresponds to
a pressure gradient parameter (λθ ) of approximately 0.06. Since the majority of
experimental data on transition in favourable pressure gradients falls within that
range (see for example [1]) the relative error between momentum thickness and
strain-rate Reynolds number is not of great concern under those conditions.
For strong adverse pressure gradients, the difference between the momentum
thickness and strain-rate Reynolds number can become significant, particularly near
separation (H = 3.5). This could either be accounted for by relating the proportionality factor to the local pressure gradient, or by including the effect directly into the
correlations. The second approach is chosen in the current model.
3. The γ -Reθ Transition Model
The main requirement for the transition model development was that only local
variables and gradients, as well as the wall distance should be used in the equations.
The wall distance can be computed from a Poisson equation and therefore does
not break the paradigm of modern CFD methods. The present formulation avoids
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283
another very severe shortcoming of the correlation-based models, namely their
limitation to 2D flows. Already the definition of a momentum-thickness is strictly
a 2D concept. It cannot be computed in general 3D flows such as a turbine blade
with sidewall boundary layers. The current formulation avoids this shortcoming, and
allows the simulation of 3D flows originating from different walls.
In the following section, a transport equation for the intermittency, γ , will be
described in detail, which can be used to trigger transition locally. The intermittency
function is coupled with the SST k-ω turbulence model [14]. It is used to turn on the
production term of the turbulent kinetic energy downstream of the transition point
in the boundary layer. This differs from the typical usage of intermittency (e.g. [34])
where the intermittency is often used to modify the eddy viscosity. However, from
a modeling standpoint, the present approach has certain advantages (e.g. capturing
the effect of large free-stream turbulence levels on laminar boundary layers and the
related increase in the laminar skin friction and heat transfer). It should be noted
that the transition model could in principle be used with a turbulence model other
than SST (e.g. k-epsilon); however, this would most likely require a recalibration
of some of the transition model constants. As well, one of the basic requirements
of the underlying turbulence model is that it must produce full turbulent flow
from the location where the model is first activated (i.e. Low-Reynolds number
turbulence models which often predict some amount of laminar flow on their own
could potentially affect the accuracy of the present transition model).
In addition to the transport equation for the intermittency, a second transport
equation is solved in terms of the transition onset momentum-thickness Reynolds
number (Reθ t ). This is necessary in order to capture the nonlocal influence of the
turbulence intensity, which changes due to the decay of the turbulent kinetic energy
in the free-stream, as well as due to changes in the free-stream velocity outside the
boundary layer. This additional transport equation is an essential part of the model
as it ties the empirical correlation to the onset criteria in the intermittency equation
and allows the model’s use in general geometries and over multiple blades, without
interaction from the user. As the transition model solves a transport equation for the
intermittency, γ , and the transitional momentum thickness Reynolds number, Reθ ,
the model was named the γ -Reθ transition model. The γ -Reθ model is one realization
of an LCTM concept.
The formulation of the intermittency equation has been extended to account
for the rapid onset of transition caused by separation of the laminar boundary
layer. In addition, the model can be fully calibrated with in-house transition onset
and transition length correlations. The correlations can also be extended to flows
at low free-stream turbulence intensity or to flows with cross-flow instability. The
model formulation therefore offers a flexible environment for engineering transition
predictions that is fully compatible with the infrastructure of modern CFD methods.
The present transition model formulation is described in five sections. The first
section details the formulation of the intermittency transport equation used to trigger
the transition onset. The second section describes the new transport equation for
the transition momentum thickness Reynolds number Reθ t , which is used to capture
the nonlocal effect of free-stream turbulence intensity and pressure gradient at the
boundary layer edge. The third section describes a modification that is used to
improve the predictions for separated flow transition. The fourth section gives an
overview of the correlations, which have to be supplied with the model. The fifth
section describes the link between the transition model and the SST model.
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3.1. Transport equation for intermittency
A new transport equation for the intermittency, γ , has been developed. The equation
is built on the one given by Menter et al. [15] but corrects most of the deficiencies of
that early formulation. A significant change to the formulation given by Menter et al.
[15] is that the intermittency is now set to be equal to one in the free stream, instead
of a small value as in the original model. This differs from the usual definition of
intermittency where the free stream intermittency is usually zero and is only equal
to one in turbulent boundary layers. However, the present approach has several
advantages, especially in stagnation regions and near the boundary layer edge, where
the original formulation did interfere with the turbulence model. The concept of a
non-zero free stream intermittency has previously been employed by Steelant and
Dick [30] in their intermittency transport equation. Although physical arguments can
be made for this modification (at least for bypass transition, where the turbulence is
diffused into the boundary layer from high free stream levels) it is mainly used here
to extend the applicability and robustness of the current method. In addition, the
empirical correlations for transition onset are applicable only for predicting boundary
layer transition, not free shear transition (i.e. a laminar jet transitioning to a turbulent
jet). As a result, away from walls, the transition model should enforce the fully
turbulent behaviour of the turbulence model.
The intermittency equation is formulated as follows:
∂(ργ ) ∂ ρU jγ
µt ∂γ
∂
+
µ+
(3)
= Pγ − Eγ +
∂t
∂xj
∂xj
σ f ∂xj
The transition source term is defined as:
0.5
Pγ 1 = Flength ca1 ρ S γ Fonset (1 − γ )
(4)
where S is the strain rate magnitude. This term is designed to be equal to zero (due to
the Fonset function) in the laminar boundary layer upstream of transition and active
wherever the local strain-rate Reynolds number exceeds the local transition onset
criteria. The magnitude of this source term is controlled by the transition length
function (Flength ). The last term in Equation (4) is used to limit the maximum value
of the intermittency so that it cannot exceed a value of one.
One of the main differences to other intermittency models lies in the formulation
of the function Fonset , which is used to trigger the intermittency production (i.e.
activate Equation 4). It is designed to switch rapidly from a value of zero in a laminar
boundary layer to a value of one downstream of transition onset. It is formulated as
a function of the local strain-rate Reynolds number Rev and the turbulent Reynolds
number ReT :
ReV =
Fonset1 =
ρy2 S
;
µ
RT =
ρk
µω
Rev
2.193 · Reθc
4
Fonset2 = min max Fonset1 , Fonset1
, 2.0
(5)
(6)
(7)
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285
Fonset3 = max 1 −
RT
2.5
3
!
,0
Fonset = max (Fonset2 − Fonset3 , 0)
(8)
(9)
One of the problems observed with the Fonset2 function was that during the
transition process the strain-rate Reynolds number would actually decrease due to
the change in the velocity profile. This could cause the transition process to stall. As
a result, the viscosity ratio was introduced in Equation (8) in order to help ensure
that the Fonset function was active throughout the transitional region. The rational
behind this equation is given in more detail in Menter et al. [15]. Reθ c in Equation (6)
is the critical Reynolds number where the intermittency first starts to increase in
the boundary layer. This occurs upstream of the transition Reynolds number, Reθt
because there is a delay due to the fact that the turbulence must first build up to
appreciable levels in the boundary layer before any change in the laminar profile can
occur. As a result, Reθc can be thought of as the location where turbulence starts to
grow while Reθt is the location where the velocity profile first starts to deviate from
the purely laminar profile. The connection between the two must be obtained from
an empirical correlation where:
Reθ c = f Reθ t
(10)
and Reθt comes from the transport equation given by Equation (16). This correlation
is determined based on a series of numerical experiments on a flat plate where the
critical Reynolds number was varied along with the free stream turbulence intensity
and the subsequent transition Reynolds number was measured based on the most
upstream location where the skin friction started to increase.
Flength in Equation (4) is an empirical correlation that controls the length of the
transition region. It is based on a significant amount of numerical experimentation
whereby a series of flat plate experiments where reproduced and a curve fitting
program was used to develop a correlation that resulted in the correct prediction
of the transition length as compared to experimental data. Once the correct values
for Flength where determined (as a function of transition Reynolds number) these
values where used to develop a correlation so that the transition length could be
correctly reproduced over the whole range of Reynolds numbers. It should be noted
that the Reθ c and Flength correlations are strong functions of each other and there was
a significant amount of iteration required in order to obtain good agreement between
both correlations. The Flength correlation is also defined as a function of Reθ t :
Flength = f Reθ t
(11)
The destruction/relaminarization source is defined as follows:
Eγ = ca2 ργ Fturb (ce2 γ − 1)
(12)
where  is the vorticity magnitude. This term acts like a sink term and ensures that
the intermittency remains close to zero (1/ce2 ) in the laminar boundary layer [it is one
in the free stream due to the inlet boundary condition and also due the presence of
wall distance in Equation (6)]. It also enables the model to predict relaminarisation
because it provides a means for the intermittency to return to zero, once the transition
criteria in the Fonset function is no longer satisfied (e.g. an accelerated boundary layer
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Flow Turbulence Combust (2006) 77: 277–303
that becomes thin enough to relaminarise). The constant ca2 controls the strength of
the destruction term and ensures that the entire term is smaller then the transition
source term Pγ . This allows the transition source term to overwhelm the destruction
term once the onset criteria is satisfied. The constant ce2 controls the lower limit of
intermittency, where the term changes sign. The value of 50 results in a lower limit
of 0.02, which is small enough to keep the boundary layer laminar. The vorticity was
used in the destruction term in order of avoiding the destruction of intermittency in
the free stream due to free-stream strain rates.
Fturb is used to disable the destruction/relaminarization source in the fully turbulent regime:
−
Fturb = e
RT
4
4
(13)
The constants for the intermittency equation are:
ca1 = 2.0;
ce2 = 50;
ca2 = 0.06;
σ f = 1.0;
(14)
The boundary condition for γ at a wall is zero normal flux. At an inlet, the value
of γ is equal to 1. In order to capture the laminar and transitional boundary layers
correctly, the grid must have a y+ of approximately 1. If the y+ is too large (i.e.
> 5) than the transition onset location moves upstream with increasing y+ . It has
also been determined that the transition onset location is sensitive to the advection
scheme used for the turbulence and transition model equations. For this reason all
equations were solved with a bounded high resolution (i.e. a limited second order)
upwind scheme. This resulted in grid independent solutions on reasonable sized grids
(e.g. for a turbine blade, 200 nodes around the blade, y+ of 1 with a wall normal grid
expansion ratio of 1.1–1.15).
3.2. Transport equation for transition momentum thickness Reynolds number
The experimental transition correlations relate the Reynolds number of transition
onset, Reθt , to the turbulence intensity, Tu, and other quantities in the free stream
where:
Reθt = f (Tu, ...) f reestream
(15)
This is a nonlocal operation, as the value of Reθt is required by the intermittency
equation inside the boundary layer, and not only in the free stream. It should also
be noted that the turbulence intensity can change strongly in a domain and that
one global value over the entire flowfield is therefore not acceptable. Examples of
such flows are highly loaded transonic turbomachinery or unsteady-state rotor-stator
interactions. Since the main requirement for the current transition model is that only
local quantities can be used, there must be a means of passing the information about
the free-stream conditions into the boundary layer.
The following transport equation can be used to resolve this issue using a local formulation. The basic concept is to treat the transition momentum thickness Reynolds
number, Reθ t , as a transported scalar quantity. The idea is then to use an empirical
correlation to calculate Reθ t in the free stream and to allow the free stream value to
diffuse into the boundary layer. This is possible because the empirical correlations are
defined as Reθt = f(Tu, dp/ds), and since Tu and dp/ds (streamwise pressure gradient)
are defined in the free stream, Reθ t is the only unknown quantity in the equation. This
Flow Turbulence Combust (2006) 77: 277–303
287
transport equation essentially takes a nonlocal empirical correlation and transforms it
into a local quantity, which can then be used to compute the transition length [Flength ,
Equation (11)] and critical Reynolds number [Reθ c , Equation (10)] at every location
in the flow field. The intermittency equation [Equation (3)] can then be solved like a
regular transport equation because the source terms are now local.
The transport equation for the transition momentum thickness Reynolds number
(Reθ t ) is defined as follows:
∂ ρU jReθ t
∂ ρReθt
∂Reθ t
∂
+
σθ t (µ + µt )
(16)
= Pθ t +
∂t
∂xj
∂xj
∂xj
Outside the boundary layer, the source term Pθ t is designed to force the transported scalar Reθ t to match the local value of Reθ t calculated from an empirical
correlation. The source term is defined as follows:
ρ
(17)
Pθ t = cθt Reθt −Reθ t (1.0 − Fθ t )
t
t=
500µ
ρU 2
(18)
where t is a time scale, which is present for dimensional reasons. The time scale was
determined based on dimensional analysis with the main criteria being that it had to
scale with the convective and diffusive terms in the transport equation. The blending
function Fθ t is used to turn off the source term in the boundary layer and allows the
transported scalar Reθt to diffuse in from the free stream. Fθt is equal to zero in the
free stream and one in the boundary layer. The Fθ t blending function is defined as
follows:
!
!
4
γ − 1/ce2 2
−( δy )
Fθt = min max Fwake · e
, 1.0
(19)
, 1.0 −
1.0 − 1/ce2
θ BL =
Reθt µ
ρU
δ BL =
15
θ BL
2
Reω =
ρωy2
µ
−
Fwake = e
Reω
1×105
δ=
50y
· δ BL
U
(20)
(21)
2
(22)
The Fwake function ensures that the blending function is not active in the wake
regions downstream of an airfoil/blade. The boundary condition for Reθt at a wall is
zero flux. The boundary condition for Reθ t at an inlet is calculated from the empirical
correlation based on the inlet turbulence intensity.
The model constants for the transport equation are as follows, where cθ t controls
the magnitude of the source term and σ θ t controls the diffusion coefficient:
cθ t = 0.03;
σθt = 2.0
(23)
There is some lag between changes in the free-stream value of Reθt and that inside the
boundary layer. The lag is desirable, as according to Abu–Ghannam and Shaw [1] the
onset of transition is primarily affected by the past history of pressure gradient and
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Flow Turbulence Combust (2006) 77: 277–303
Figure 3 Results for flat plate test cases with different freestream turbulence levels (FSTI –
Freestream Turbulence Intensity).
turbulence intensity and not the local value at transition. The lag between the local
value of Reθ t in the boundary layer and that in the free stream can be controlled by
the diffusion coefficient σ θ t . The larger the value of σ θ t the less sensitive the transition
model is to history effects. The present value of σ θ t = 2 was obtained based on the
T3 series of flat plate transition experiments (Figures 3 and 4) were the free stream
turbulence intensity and pressure gradient were rapidly changing and flow history
effects are believed to be significant.
3.3. Separation induced transition
It became apparent during the development of the present transition model that
whenever a laminar boundary layer separation occurred, the model consistently
predicted the turbulent reattachment location too far downstream. Based on experimental results for a low-pressure turbine blade the agreement with experimental data
tended to decrease as the free stream turbulence intensity was lowered. Presumably
this is because the turbulent kinetic energy, k, in the separating shear layer is smaller
at lower free stream turbulence intensities. As a result, it takes longer for k to grow
to large enough values that would cause the boundary layer to reattach. This is the
case even if the onset of transition is predicted at or near the separation point.
To correct this deficiency, a modification to the transition model was introduced
that allows k to grow rapidly once the laminar boundary layer separates. The
modification has been formulated so that it has a negligible effect on the predictions
for attached transition or fully turbulent flow. The main idea behind the separation-
Flow Turbulence Combust (2006) 77: 277–303
289
Figure 4 Results for flat plate test cases where variation of the tunnel Reynolds number causes
transition to occur in different pressure gradients (dp/dx).
induced transition correction is to allow the local intermittency to exceed 1 whenever
the laminar boundary layer separates. This will result in a large production of k,
which in turn will cause earlier reattachment. The means for accomplishing this is
based on the fact that for a laminar separation the strain-rate Reynolds number
Rev significantly exceeds the critical momentum thickness Reynolds number Reθ c .
As a result, the ratio between the two (when Rev > Reθ c ) can be thought of as a
measure of the size of the laminar separation and can therefore be used to increase
the production of turbulent kinetic energy. The modification for separation-induced
transition is given by:
Rev
γsep = min s1 max 0,
− 1 Freattach , 2 Fθt (24)
3.235 Reθ c
Freattach = e
4
R
− 20T
γef f = max γ , γsep
(25)
(26)
s1 = 2
The size of the separation bubble can be controlled with the constant s1 . Freattach disables the modification once the viscosity ratio is large enough to cause reattachment.
Fθ t is the blending function from the Reθ t equation [Equation (16)] and confines the
modification to boundary layer flows. The constant 3.235 is the relation between Rev
and Reθ for a shape factor (H) equal to 3.5 (i.e. at the separation point, see Figure 2).
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Flow Turbulence Combust (2006) 77: 277–303
The ability to include such a complex effect as separation-induced transition in such
a simple way into the model shows the flexibility of the current approach.
3.4. Empirical correlations
The model contains three empirical correlations. Reθ t is the transition onset as
observed in experiments and it is used in Equation (17). Flength is the length of the
transition zone and goes into Equation (4). Reθ c is the point where the model is
activated in order to match both, Reθt and Flength , and it goes into Equation (6).
(27)
Reθ t = f (Tu, λθ ); Flength = f Reθ t ; Reθ c = f Reθ t
The turbulence intensity, Tu, and Thwaites’ pressure gradient coefficient, λθ , are
defined as:
p
2k/3
ρθ 2 dU
Tu = 100
; λθ =
(28)
U
µ ds
At present these empirical correlations are proprietary and cannot be given in the
paper.
3.5. Coupling the transition model with the turbulence model
The Reynolds stresses and turbulent heat fluxes in the Navier–Stokes and energy
equations are modeled using the eddy-viscosity/diffusivity approach. The eddy diffusivity in the energy equation is related to the eddy viscosity via the turbulent Prandtl
number (Prt = 0.9). The eddy viscosity (µt ) is calculated from the following modified
version of the SST turbulence model [14] as follows:
ρk a1 ρk
µt = min
;
(29)
ω SF2
∂
∂
∂k
∂
(ρk) +
(ρu jk) = P̃k − D̃k +
(µ + σk µt )
∂t
∂xj
∂xj
∂xj
∂
∂
∂ω
Pk
∂
(ρω) +
(ρu jω) = α
− Dω + Cdω +
(µ + σk µt )
∂t
∂xj
νt
∂xj
∂xj
P̃k = γef f Pk ;
D̃k = min max(γef f , 0.1), 1.0 Dk
(30)
(31)
(32)
where Pk and Dk are the production and destruction terms from the turbulent kinetic
energy equation. The only difference between these equations and the original SST
model is the appearance of the effective intermittency [γ eff , see Equation (26)] in
Equation (32). It should be stressed that the intermittency is used only to control
the source terms in the k-equation, it is not used to multiply the eddy viscosity
[Equation (29)]. In this way the present concept of intermittency is somewhat
different than the standard definition used by Steelant and Dick [30] or Suzen et
al. [32]. However, the present approach has two main advantages. The first is that
it improves the robustness of the model because the intermittency does not enter
directly into the momentum equations. The second advantage is that it allows the
model to predict the effects of high free stream turbulence levels on buffeted laminar
Flow Turbulence Combust (2006) 77: 277–303
291
boundary layers. The reason is that for large free stream eddy viscosities, the small
values of intermittency in the boundary layer do not cancel out the local eddy
viscosity. Consequently, the increase in the laminar shear stress and heat transfer
that has been observed experimentally in buffeted laminar boundary layers can in
principle, be captured by the present model.
One further modification to the SST turbulence model is a change in the blending
function F1 responsible for switching between the k-ω and k-ε models. It was found
that in the center of the laminar boundary layer F1 could potentially switch from
1.0 to 0.0. This is not desirable, as the k-ω model must be active in the laminar and
transitional boundary layers. The deficiency in the blending function is not surprising
as the equations used to define F1 were intended solely for use in turbulent boundary
layers. The solution is to redefine F1 in terms of a blending function that will always
be equal to 1.0 in a laminar boundary layer. The modified blending function is defined
as follows:
√
ρy k
Ry =
(33)
µ
−
F3 = e
Ry
120
8
F1 = max F1orig , F3
(34)
(35)
where F1orig is the original blending function from the SST turbulence model.
4. Test Cases
The remaining part of the paper will give an overview of some of the public-domain
test cases which have been computed with the model described above. This naturally
requires a compact representation of the simulations. The cases are described in more
detail in Menter et al. [16], Langtry et al. [6] and Langtry and Menter [7], including
grid refinement and sensitivity studies.
4.1. Flat plate test cases
Test cases presented are the ERCOFTAC T3 series of flat plate experiments [20,
21, 22] and the Schubauer and Klebanof [24] flat plate experiment, all of which are
commonly used as benchmarks for transition models. Also included is a test case
where the boundary layer experiences a strong favourable pressure gradient that
causes it to relaminarise [13].
The three cases (T3A, T3A and T3B) have zero pressure gradients with different
free-stream turbulence intensity (FSTI) levels corresponding to transition in the
bypass regime. The Schubauer and Klebanof (S&K) test case has a low freestream turbulence intensity and corresponds to natural transition. Figure 3 shows the
comparison of the model prediction with experimental data for theses cases. It also
gives the corresponding FSTI values. In all simulations, the inlet turbulence levels
were specified to match the experimental turbulence intensity and its decay rate. As
the free-stream turbulence increases, the transition location moves to lower Reynolds
numbers.
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Flow Turbulence Combust (2006) 77: 277–303
Figure 5 Predicted skin
friction (Cf) for a flat plate
with a strong acceleration that
causes the boundary layer to
relaminarize.
The T3C test cases consist of a flat plate with a favourable and adverse pressure gradient imposed by the opposite converging/diverging wall. The wind tunnel
Reynolds number was varied for the four cases (T3C5, T3C3, T3C2, T3C4) thus
moving the transition location from the favourable pressure at the beginning of the
plate to the adverse pressure gradient at the end. The cases are used to demonstrate
the transition models ability to predict transition under the influence of various pressure gradients. Figure 4 details the results for the pressure gradient cases. The effect
of the pressure gradient on the transition length is clearly visible with favourable
pressure gradients increasing the transition length and adverse pressure gradients
reducing it. For the T3C4 case the laminar boundary layer actually separates and
undergoes separation induced transition.
The relaminarisation test case is shown in Figure 5. For this case the opposite
converging wall imposes a strong favourable pressure gradient that can relaminarise
a turbulent boundary layer. In both the experiment and in the CFD prediction the
boundary layer was tripped near the plate leading edge. In the CFD computation
this was accomplished by injecting a small amount of turbulent air into the boundary
layer. Downstream of the trip the boundary layer slowly relaminarises due to the
strong favourable pressure gradient.
For all of the flat plate test cases the agreement with the data is generally
good, considering the diverse nature of the physical phenomena computed, ranging
from bypass transition to natural transition, separation-induced transition and even
relaminarisation. Note that such a wide range of transition mechanisms is outside the
scope of any other available transition model.
4.2. Turbomachinery test cases
4.2.1. Zierke & Deutsch compressor cascade
For the present test case [40], transition on the suction side occurs at the leading edge
due to a small leading edge separation bubble on the suction side. On the pressure
side, transition occurs at about mid-chord. The turbulence contours and the skin
friction distribution are shown in Figure 6. There appears to be a significant amount
of scatter in the experimental data, however, overall the transition model is predicting
Flow Turbulence Combust (2006) 77: 277–303
293
Figure 6 Turbulence intensity
contours (top) and
cf-distribution against
experimental data (right)
for the Zierke & Deutsch
compressor.
the major flow features correctly (i.e. fully turbulent suction side, transition at midchord on the pressure side). One important issue to note is the effect of streamwise grid resolution on resolving the leading edge laminar separation and subsequent
transition on the suction side. If the number of stream-wise nodes clustered around
the leading edge is too low, the model cannot resolve the rapid transition and a
laminar boundary layer on the suction side is the result. For the present study, 60
streamwise nodes were used between the leading edge and the x/C equal to 0.1
location.
4.2.2. Von-Karman institute turbine cascade
The surface heat transfer for the transonic VKI MUR 241 (FSTI = 6.0%) and MUR
116 (FSTI = 1.0%) test cases [2] is shown in Figure 7. The strong acceleration on
the suction side for the MUR 241 case keeps the flow laminar until a weak shock
at mid chord, whereas for the MUR 116 case the flow is laminar until right before
the trailing edge. Downstream of transition there appears to be a significant amount
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Flow Turbulence Combust (2006) 77: 277–303
Figure 7 Heat transfer for the
VKI MUR241 (FSTI = 6.0%)
and MUR116 (FSTI = 1.0%)
test cases.
of error between the predicted turbulent heat transfer and the measured value. It is
possible that this is the result of a Mach number (inlet Mach number Mainlet = 0.15,
Maoutlet = 1.089) effect on the transition length [30, 31]. At present, no attempt has
been made to account for this effect in the model. It can be incorporated in future
correlations, if found consistently important.
The pressure side heat transfer is of particular interest for this case. For both
cases, transition did not occur on the pressure side, however, the heat transfer was
significantly increased for the high turbulence intensity case. This is a result of the
large free-stream levels of turbulence which diffuse into the laminar boundary layer
and increase the heat transfer and skin friction. From a modeling standpoint, the
effect was caused by the large free-stream viscosity ratio necessary for MUR 241
to keep the turbulence intensity from decaying below 6%, which is the free-stream
value quoted in the experiment. The enhanced heat transfer on the pressure side was
also present in the experiment and the effect appears to be physical. The model can
predict this effect, as the intermittency does not multiply the eddy-viscosity but only
the production term of the k-equation. The diffusive terms are therefore active in the
laminar region.
4.2.3. RGW compressor cascade
The RGW annular compressor [25] features a fully three-dimensional flow, including
sidewall boundary layers originating upstream of the blade. This flow topology poses
a major challenge to standard correlation-based transition models, as complex logic
would be required to distinguish between the different boundary layers.
Figure 8 shows a comparison of the simulations on the suction side of the blade
with an experimental oil-flow picture. For comparison, a fully turbulent flow simulation is also included. It can be seen that the transition model captures the complex
flow topology of the experiments in good agreement with the data. A comparison
between the transition model and the fully turbulent simulation shows the strong
influence of the laminar flow separation on the sidewall boundary layer separation.
The flow separation on the shroud is significantly reduced by the displacement effect
of the separation bubble in the transitional simulation. As a result, the loss coefficient,
Yp = 0.19, in the fully turbulent simulation is much higher than the experimental
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295
Figure 8 Fully turbulent (left) and transitional (right) skin friction on the suction side of the 3D RGW
compressor cascade compared to experimental oil flow visualization (middle). (Yp – loss coefficient).
value of Yp = 0.097. The simulation with the transition model gives a value of
Yp = 0.11 in much closer agreement with the experiment. More turbomachineryrelated test cases can be found in Langtry et al. [6], including an unsteady rotor-stator
interaction simulation.
4.3. Wind turbine test case
The test case geometry is a 2D airfoil section, as typically used for GE wind-turbine
blades. It operates in a low FSTI environment with a turbulence intensity of only
around 0.1% at the leading edge. As a result, natural transition occurs on both the
suction and pressure surfaces. The inlet value for the ω in this application was chosen
to match the experimental transition location at 0◦ angle of attack. All other angles
of attack have been computed with the identical settings. For a detailed discussion,
see Langtry et al. [5].
The transition locations vs. angle of attack for the present transition model are
shown in Figure 9 (left). Wind tunnel results and predictions XFOIL (v6.8) based on
an en method are plotted for comparison. The experimental data were obtained using
Figure 9 Predicted transition location (left) and drag coefficient (right) as a function of angle of attack
for a wind turbine airfoil in comparison with experiments and the en -method used in XFOIL.
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Flow Turbulence Combust (2006) 77: 277–303
a stethoscope method. The current model captures the dependence of the transition
location on the angle of attack in very good agreement with the data. The effect of
the transition model is clearly visible also in the drag coefficient Figure 9 (right).
Numbers could not be provided on the y axis, due to data confidentiality.
4.3.1. 3D NREL wind turbine simulation
Simulations have been carried out for the NREL wind turbine [26]. This is a
notoriously difficult test case to compute with CFD and a detailed comparison of
CFD and experimental data is beyond the scope of this paper. However, in the
simulation of this flow, it turned out that some interesting differences were observed
between fully turbulent and transitional results at severe stall conditions.
The suction side intermittency predicted by the transitional CFD computations
for wind speeds of 7, 10 and 20 m/s are shown in Figure 10. As the wind speed is
increased the effective angle of attack of the wind turbine increases. For the 7 m/s
wind speed the flow is largely attached on the suction side. In addition, a significant
amount of laminar flow is predicted near the tip as well as in the hub region. In the tip
region transition occurs at the 0.5 chord position whereas in the middle of the blade
span transition occurs near the leading edge. This is most likely caused by the smaller
radius resulting in an increased effective angle of attack. For the 10 m/s case, the
inner hub region stalls while the tip region remains attached up until the 0.5 chord
position. As well, due to the increased angle of attack the transition location near
the tip moves to the leading edge. Finally, at 20 m/s the suction side of the blade is
completely separated and the intermittency contours indicate that the flow is almost
completely turbulent.
Figure 11 shows the shaft torque in comparison with experiments (left) and the
flow topologies computed for fully turbulent and transitional settings. At a wind
speed of 20 m/s, the flow topology computed with the fully turbulent and the
transitional approaches are very different. This results in an 80% change in output
torque. The lower output torque appears to be the result of a laminar separation in
the leading edge region of the suction side of the blade. The transitional simulation is
in much closer agreement with the experimental data.
Figure 10 NREL wind
turbine, suction surface
intermittency computed by the
transition model for wind
speeds of 7 m/s (top), 10 m/s
(middle) and 20 m/s (bottom).
Flow Turbulence Combust (2006) 77: 277–303
297
Figure 11 Shaft torque at different wind speeds (left). Flow topology on suction side for fully
turbulent and transitional simulations (right).
4.3.2. Aeronautical test cases
Transition in aeronautical flows is typically a result of Tollmien–Schlichting waves
or a crossflow instability. The current model does presently not include correlations
for crossflow instabilities. It does however account for natural transition including
pressure gradients. For more details on the test cases see Langtry and Menter [7].
4.3.3. McDonald Douglas 30P–30N f lap
The McDonald Douglas 30P–30N flap configuration was originally a test case for the
High-Lift Workshop/CFD Challenge that was held at the NASA Langley Research
Center in 1993 [4]. It is a very complex test case for a transition model because of
the large changes in pressure gradient and the varying local free-stream turbulence
intensity around the different lifting surfaces.
The experiment was performed in the Langley Low Turbulence Pressure Tunnel
and the transition locations were measured using hot films on the upper surface of
the slat and flap and on both the upper and lower surfaces of the main element. The
skin friction was also measured at various locations using a Preston tube [4]. For the
present comparison the Reynolds number Re = 9 × 106 and an angle of attack α = 8◦
was selected. The free-stream conditions for k and ω were selected to match the
transition location at the suction side of the slat. The other transition locations are
an outcome of the simulation.
A contour plot of the predicted turbulence intensity around the flap is shown in
Figure 12. Also indicated are the various transition locations that were measured in
the experiment (Exp.) as well as the locations predicted by the present transition
model (CFD). In the computations, the onset of transition was judged as the location
were the skin friction first started to increase due to the production of turbulent
kinetic energy in the boundary layer. In general the agreement between the measured
and predicted transition locations is very good. The largest error was observed on the
lower surface of the main element where the predicted transition location was too far
downstream by approximately 6% of the cruise-airfoil chord.
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Flow Turbulence Combust (2006) 77: 277–303
Figure 12 Contour of turbulence intensity (Tu) around the McDonald Douglas 30P-30N flap as well
as the measured (Exp.) and predicted (CFD) transition locations (x/c) as a function of the cruiseairfoil chord (c = 0.5588 m). Also indicated is the relative error between the experiment and the
predictions.
4.3.4. DLR F-5 wing
The DLR F-5 geometry is a 20◦ swept wing with a symmetrical airfoil section that is
supercritical at a free-stream Mach number of 0.82. The experiment was performed
at the DLR by Sobieczky [28] and consists of a wing mounted to the tunnel sidewall
(which is assumed to have transitioned far upstream of the wing). At the root the
wing was designed to blend smoothly into the wall thus eliminating the horseshoe
vortex that usually develops there. The experimental measurements consist of wing
mounted static taps at various span wise locations and flow visualization of the surface
shear using a sublimation technique. The experimental flow visualization is shown in
Figure 13 (right). Based on the flow visualization and the pressure measurements
a diagram of the flow field around the wing was constructed and can be seen in
Figure 13 (middle). From the measurements the boundary layer is laminar until
about 60% chord where a shock causes the laminar boundary layer to separate and
reattach as a turbulent boundary layer. The contours of skin friction and the surface
streamlines predicted by the transition model are shown in Figure 13 (left). From
the skin friction the laminar separation and turbulent reattachment can be clearly
seen and both appear to be in good agreement with the experimental diagram from
about 20% span out to the wing tip. Near the wing-body intersection, the experiments
indicate earlier transition than the simulations. This might be due to the omission of
the crossflow instability in the transition model.
4.3.5. Eurocopter cabin
The helicopter test case was investigated to demonstrate the model’s ability of solving
flows around complex geometries including multiple physical effects. There are no
experimental data available in the public domain, which precludes the use of the
Flow Turbulence Combust (2006) 77: 277–303
299
Figure 13 DLR-F5 wing with transition. Simulations (left); experiment (middle and right). Note:
What is labeled a recompression shock in the experiment is most likely due to the strong pressure
change associated with a reattaching boundary layer.
test case for model validation. Nevertheless, interesting transitional phenomena can
be observed. In addition, the numerical performance of the transition model can be
investigated.
The grid for this case consisted of about 6 million nodes and each solution was
run overnight in parallel on a 16 CPU 1.5 Ghz Linux cluster. The convergence of
lift and drag is shown in Figure 14 for the fully turbulent (top) and transitional
(bottom) solutions. The transition model does not appear to have any adverse effects
on the convergence and converges similar to the fully turbulent solution. This was
Figure 14 Convergence of lift and drag for the fully turbulent (left) and transitional (right) Eurocopter simulation.
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Flow Turbulence Combust (2006) 77: 277–303
Figure 15 Contour plot of skin friction for a fully turbulent (left) and transitional (right) solution of
the Eurocopter cabin. Isosurface indicates reverse flow.
also observed in most of the other cases. Typically the convergence is somewhat
reduced in the transitional simulations, as the transition location has to settle down
before convergence can be reached. The overall increase (additional equations and
convergence) of the model is typically ∼20%. The transitional flow on the fuselage
and tails resulted in a 5% drag reduction compared to the fully turbulent solution.
The predicted skin friction for a fully turbulent and transitional solution is shown
in Figure 15. The main differences in the transitional solution are that the front
part of the fuselage, the two outside vertical tail surfaces and the outer half of
the horizontal tail surface are laminar. The fact that the transition model predicted
turbulent flow on the middle vertical stabilizer and the inner part of the horizontal
stabilizer was unexpected. Further investigation revealed that this was caused by the
turbulent wake that was shed from the fuselage upstream of the tail. This is best
illustrated in Figure 16. The left picture shows an iso-surface of the turbulent flow.
The turbulent wake is clearly visible and can be seen passing over the middle vertical
stabilizer and the inner part of the horizontal stabilizer. Consequently, the transition
Figure 16 Iso-surface of turbulent flow (left) and surface value of intermittency (right) indicating the
laminar (blue) and turbulent (red) regions on the Eurocopter airframe.
Flow Turbulence Combust (2006) 77: 277–303
301
model predicts bypass transition on these surfaces due to the high local free-stream
turbulence intensity from the wake.
The Eurocopter test case demonstrates the potential of the transition model
for solving complex aerodynamic flow problems, with the inclusion of first order
transitional effects. Further model refinements are required for calibration of the
model for such flows, including a model extension for crossflow instability.
5. Conclusions
Methods for transition prediction in general purpose CFD codes have been discussed.
The requirements, which a model has to satisfy to be suitable for implementation into
such a code, have been listed. The main criterion is that nonlocal operations must
be avoided. A new concept of transition modeling termed Local Correlation-based
Transition Model (LCTM) was introduced. It combines the advantages of locally
formulated transport equations with the physical information contained in empirical
correlations. The γ -Reθ transition model is a representative of that modeling concept.
The model is based on two transport equations, one for intermittency and one
for a transition onset criterion in terms of momentum thickness Reynolds number.
The proposed transport equations do not attempt to model the physics of the
transition process (unlike, e.g., turbulence models), but form a framework for the
implementation of transition correlations into general-purpose CFD methods.
An overview of the γ -Reθ model formulation has been given and numerous test
cases have been computed successfully. The authors believe that the current model is
a significant step forward in engineering transition modeling. Because it is based on
transport equations, the model formulation therefore offers a flexible environment
for engineering transition predictions that is fully compatible with the infrastructure
of modern CFD methods. As a result, the model can be used in any general purpose
CFD method without special provisions for geometry and grid topology.
The present transition model accounts for transition due to free-stream turbulence
intensity, pressure gradients and separation. It is fully CFD-compatible and does
not negatively affect the convergence of the solver. Current limitations of the
model are that crossflow instability is not included in the correlations and that the
transition correlations are formulated non-Galilean invariant. Both limitations are
currently being investigated and can be removed in principle. Another area for future
investigations is the important effect of wake induced transition, as it occurs on blades
of axial turbines/compressors affected by the wake of upstream blade rows. The
helicopter test case shows that wake-induced transition can be handled by the model.
The increased FSTI caused by the wake, affects the transition onset correlation,
leading to earlier transition. Tests for unsteady turbomachinery cases are required
for further validation of this functionality.
An overview of the test cases computed with the new model has been given. Due
to the nature of the paper, the presentation of each individual test case had to be
brief. More details on the test case set-up, boundary conditions grid resolutions etc.
can be found in the cited papers. The purpose of the overview was to show that the
model can handle a wide variety of geometries and physically diverse problems. The
authors believe that the LCTM concept of combining transition correlations with
locally formulated transport equations has a strong potential for allowing the 1st
order effects of transition to be included into today’s industrial CFD simulations.
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Acknowledgement The model development and validation at ANSYS CFX was funded by GE
Aircraft Engines and GE Global Research. Prof. G. Huang and Dr B. Suzen from the University of
Kentucky have supported the original model development with their extensive know-how and their
in-house codes.
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